Ergodic theory of nonlinear markov operators with applications in nonlinear consensus problems

A consensus problem of multi-agent systems has been considered a dual problem to Markov chains. Historically, an idea of reaching consensus through linear repeated averaging was introduced by DeGroot (see [1]) for a structured time-invariant and synchronous environment. Since that time, the consensus which is the most ubiquitous phenomenon of multi-agent systems becomes popular in various scientific communities, such as biology, physics, control engineering and social science. In [2], Chatterjee and Seneta considered a generalization of DeGroot’s model for the structured time-varying synchronous environment. Based on the ergodic theory of non-homogeneous Markov chains, the theory of linear consensus problems for structured time-varying environment was established very well [2]. In this plenary talk, we shall present the ergodic theory of nonlinear Markov chains which enables to study the nonlinear consensus problems for the structured time-invariant as well as time-varying synchronous environment. This theory was developed in the series of papers [3, 4, 5, 6]. This work was supported by Ministry of Higher Education (MOHE) grant FRGS14-141-0382